Cubic critical

Percentage Accurate: 52.3% → 85.5%

Time: 12.1s

Alternatives: 14

Speedup: 11.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+154)
   (/ 1.0 (* -1.5 (/ a b)))
   (if (<= b 2.35e-84)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+154) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 2.35e-84) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d+154)) then
        tmp = 1.0d0 / ((-1.5d0) * (a / b))
    else if (b <= 2.35d-84) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+154) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 2.35e-84) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e+154:
		tmp = 1.0 / (-1.5 * (a / b))
	elif b <= 2.35e-84:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+154)
		tmp = Float64(1.0 / Float64(-1.5 * Float64(a / b)));
	elseif (b <= 2.35e-84)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e+154)
		tmp = 1.0 / (-1.5 * (a / b));
	elseif (b <= 2.35e-84)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+154], N[(1.0 / N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.35e-84], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3e154

   1. Initial program 54.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 95.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   7. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   8. Step-by-step derivation

      1. associate-*l/ 95.2%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   10. Step-by-step derivation

       1. clear-num 95.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

       2. inv-pow 95.3%

          \[\leadsto \color{blue}{{\left(\frac{a}{b \cdot -0.6666666666666666}\right)}^{-1}} \]

       3. *-un-lft-identity 95.3%

          \[\leadsto {\left(\frac{\color{blue}{1 \cdot a}}{b \cdot -0.6666666666666666}\right)}^{-1} \]

       4. *-commutative 95.3%

          \[\leadsto {\left(\frac{1 \cdot a}{\color{blue}{-0.6666666666666666 \cdot b}}\right)}^{-1} \]

       5. times-frac 95.4%

          \[\leadsto {\color{blue}{\left(\frac{1}{-0.6666666666666666} \cdot \frac{a}{b}\right)}}^{-1} \]

       6. metadata-eval 95.4%

          \[\leadsto {\left(\color{blue}{-1.5} \cdot \frac{a}{b}\right)}^{-1} \]

   11. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{{\left(-1.5 \cdot \frac{a}{b}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 95.4%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

   13. Simplified 95.4%

       \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

   if -2.3e154 < b < 2.35e-84

   1. Initial program 87.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 2.35e-84 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+154)
   (/ 1.0 (* -1.5 (/ a b)))
   (if (<= b 1.02e-81)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+154) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 1.02e-81) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d+154)) then
        tmp = 1.0d0 / ((-1.5d0) * (a / b))
    else if (b <= 1.02d-81) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+154) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 1.02e-81) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e+154:
		tmp = 1.0 / (-1.5 * (a / b))
	elif b <= 1.02e-81:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+154)
		tmp = Float64(1.0 / Float64(-1.5 * Float64(a / b)));
	elseif (b <= 1.02e-81)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e+154)
		tmp = 1.0 / (-1.5 * (a / b));
	elseif (b <= 1.02e-81)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+154], N[(1.0 / N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e-81], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3e154

   1. Initial program 54.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 54.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 95.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 95.2%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   7. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   8. Step-by-step derivation

      1. associate-*l/ 95.2%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   10. Step-by-step derivation

       1. clear-num 95.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

       2. inv-pow 95.3%

          \[\leadsto \color{blue}{{\left(\frac{a}{b \cdot -0.6666666666666666}\right)}^{-1}} \]

       3. *-un-lft-identity 95.3%

          \[\leadsto {\left(\frac{\color{blue}{1 \cdot a}}{b \cdot -0.6666666666666666}\right)}^{-1} \]

       4. *-commutative 95.3%

          \[\leadsto {\left(\frac{1 \cdot a}{\color{blue}{-0.6666666666666666 \cdot b}}\right)}^{-1} \]

       5. times-frac 95.4%

          \[\leadsto {\color{blue}{\left(\frac{1}{-0.6666666666666666} \cdot \frac{a}{b}\right)}}^{-1} \]

       6. metadata-eval 95.4%

          \[\leadsto {\left(\color{blue}{-1.5} \cdot \frac{a}{b}\right)}^{-1} \]

   11. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{{\left(-1.5 \cdot \frac{a}{b}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 95.4%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

   13. Simplified 95.4%

       \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

   if -2.3e154 < b < 1.01999999999999998e-81

   1. Initial program 87.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 87.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 87.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 87.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   if 1.01999999999999998e-81 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-83}:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot \left(-3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.08e-83)
     (* (- b (sqrt (* a (* c (- 3.0))))) (/ (/ 1.0 a) -3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.08e-83) {
		tmp = (b - sqrt((a * (c * -3.0)))) * ((1.0 / a) / -3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.08d-83) then
        tmp = (b - sqrt((a * (c * -3.0d0)))) * ((1.0d0 / a) / (-3.0d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.08e-83) {
		tmp = (b - Math.sqrt((a * (c * -3.0)))) * ((1.0 / a) / -3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.08e-83:
		tmp = (b - math.sqrt((a * (c * -3.0)))) * ((1.0 / a) / -3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.08e-83)
		tmp = Float64(Float64(b - sqrt(Float64(a * Float64(c * Float64(-3.0))))) * Float64(Float64(1.0 / a) / -3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.08e-83)
		tmp = (b - sqrt((a * (c * -3.0)))) * ((1.0 / a) / -3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e-83], N[(N[(b - N[Sqrt[N[(a * N[(c * (-3.0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999993e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.1999999999999993e-114 < b < 1.08e-83

   1. Initial program 78.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.5%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in c around -inf 76.4%

       \[\leadsto \left(b - \sqrt{\color{blue}{-1 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt[3]{3}\right)}^{3}\right)\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]
   12. Step-by-step derivation

       1. associate-*r* 76.4%

          \[\leadsto \left(b - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot \left(c \cdot {\left(\sqrt[3]{3}\right)}^{3}\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       2. neg-mul-1 76.4%

          \[\leadsto \left(b - \sqrt{\color{blue}{\left(-a\right)} \cdot \left(c \cdot {\left(\sqrt[3]{3}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       3. rem-cube-cbrt 76.5%

          \[\leadsto \left(b - \sqrt{\left(-a\right) \cdot \left(c \cdot \color{blue}{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   13. Simplified 76.5%

       \[\leadsto \left(b - \sqrt{\color{blue}{\left(-a\right) \cdot \left(c \cdot 3\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   if 1.08e-83 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-83}:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot \left(-3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 10^{-82}:\\ \;\;\;\;\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \frac{\frac{-1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1e-82)
     (* (sqrt (* (* a c) -3.0)) (/ (/ -1.0 a) -3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1e-82) {
		tmp = sqrt(((a * c) * -3.0)) * ((-1.0 / a) / -3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1d-82) then
        tmp = sqrt(((a * c) * (-3.0d0))) * (((-1.0d0) / a) / (-3.0d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1e-82) {
		tmp = Math.sqrt(((a * c) * -3.0)) * ((-1.0 / a) / -3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1e-82:
		tmp = math.sqrt(((a * c) * -3.0)) * ((-1.0 / a) / -3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1e-82)
		tmp = Float64(sqrt(Float64(Float64(a * c) * -3.0)) * Float64(Float64(-1.0 / a) / -3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1e-82)
		tmp = sqrt(((a * c) * -3.0)) * ((-1.0 / a) / -3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-82], N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999993e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.1999999999999993e-114 < b < 1e-82

   1. Initial program 78.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.5%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around 0 75.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   12. Step-by-step derivation

       1. mul-1-neg 75.8%

          \[\leadsto \color{blue}{\left(-\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

       2. rem-cube-cbrt 75.9%

          \[\leadsto \left(-\sqrt{a \cdot \left(c \cdot \color{blue}{-3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       3. associate-*r* 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       4. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       5. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{-3 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   13. Simplified 75.9%

       \[\leadsto \color{blue}{\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

   if 1e-82 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 10^{-82}:\\ \;\;\;\;\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \frac{\frac{-1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \frac{-0.3333333333333333}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 3.15e-83)
     (* (sqrt (* (* a c) -3.0)) (/ -0.3333333333333333 (- a)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 3.15e-83) {
		tmp = sqrt(((a * c) * -3.0)) * (-0.3333333333333333 / -a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 3.15d-83) then
        tmp = sqrt(((a * c) * (-3.0d0))) * ((-0.3333333333333333d0) / -a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 3.15e-83) {
		tmp = Math.sqrt(((a * c) * -3.0)) * (-0.3333333333333333 / -a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 3.15e-83:
		tmp = math.sqrt(((a * c) * -3.0)) * (-0.3333333333333333 / -a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 3.15e-83)
		tmp = Float64(sqrt(Float64(Float64(a * c) * -3.0)) * Float64(-0.3333333333333333 / Float64(-a)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 3.15e-83)
		tmp = sqrt(((a * c) * -3.0)) * (-0.3333333333333333 / -a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.15e-83], N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * N[(-0.3333333333333333 / (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999993e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.1999999999999993e-114 < b < 3.14999999999999983e-83

   1. Initial program 78.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.5%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around 0 75.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   12. Step-by-step derivation

       1. mul-1-neg 75.8%

          \[\leadsto \color{blue}{\left(-\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

       2. rem-cube-cbrt 75.9%

          \[\leadsto \left(-\sqrt{a \cdot \left(c \cdot \color{blue}{-3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       3. associate-*r* 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       4. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       5. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{-3 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   13. Simplified 75.9%

       \[\leadsto \color{blue}{\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

   14. Taylor expanded in a around 0 75.9%

       \[\leadsto \left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \color{blue}{\frac{-0.3333333333333333}{a}} \]

   if 3.14999999999999983e-83 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \frac{-0.3333333333333333}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.55e-84)
     (/ (/ (sqrt (* a (* c -3.0))) a) 3.0)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.55e-84) {
		tmp = (sqrt((a * (c * -3.0))) / a) / 3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.55d-84) then
        tmp = (sqrt((a * (c * (-3.0d0)))) / a) / 3.0d0
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.55e-84) {
		tmp = (Math.sqrt((a * (c * -3.0))) / a) / 3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.55e-84:
		tmp = (math.sqrt((a * (c * -3.0))) / a) / 3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.55e-84)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) / a) / 3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.55e-84)
		tmp = (sqrt((a * (c * -3.0))) / a) / 3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-84], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999993e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.1999999999999993e-114 < b < 1.55000000000000001e-84

   1. Initial program 78.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.5%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around 0 75.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   12. Step-by-step derivation

       1. mul-1-neg 75.8%

          \[\leadsto \color{blue}{\left(-\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

       2. rem-cube-cbrt 75.9%

          \[\leadsto \left(-\sqrt{a \cdot \left(c \cdot \color{blue}{-3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       3. associate-*r* 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       4. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       5. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{-3 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   13. Simplified 75.9%

       \[\leadsto \color{blue}{\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   14. Step-by-step derivation

       1. associate-*r/ 75.9%

          \[\leadsto \color{blue}{\frac{\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a}}{-3}} \]

       2. frac-2neg 75.9%

          \[\leadsto \color{blue}{\frac{-\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a}}{--3}} \]

   15. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{a \cdot \left(-3 \cdot c\right)}}{a}}{3}} \]

   if 1.55000000000000001e-84 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.7e-82) (/ (sqrt (* a (* c -3.0))) (* a 3.0)) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.7e-82) {
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.7d-82) then
        tmp = sqrt((a * (c * (-3.0d0)))) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.7e-82) {
		tmp = Math.sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.7e-82:
		tmp = math.sqrt((a * (c * -3.0))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.7e-82)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.7e-82)
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-82], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999993e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.1999999999999993e-114 < b < 1.69999999999999988e-82

   1. Initial program 78.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.5%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.4%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around 0 75.8%

       \[\leadsto \color{blue}{\left(-1 \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   12. Step-by-step derivation

       1. mul-1-neg 75.8%

          \[\leadsto \color{blue}{\left(-\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]

       2. rem-cube-cbrt 75.9%

          \[\leadsto \left(-\sqrt{a \cdot \left(c \cdot \color{blue}{-3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       3. associate-*r* 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       4. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

       5. *-commutative 75.9%

          \[\leadsto \left(-\sqrt{-3 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   13. Simplified 75.9%

       \[\leadsto \color{blue}{\left(-\sqrt{-3 \cdot \left(c \cdot a\right)}\right)} \cdot \frac{\frac{1}{a}}{-3} \]
   14. Step-by-step derivation

       1. distribute-lft-neg-out 75.9%

          \[\leadsto \color{blue}{-\sqrt{-3 \cdot \left(c \cdot a\right)} \cdot \frac{\frac{1}{a}}{-3}} \]

       2. neg-sub0 75.9%

          \[\leadsto \color{blue}{0 - \sqrt{-3 \cdot \left(c \cdot a\right)} \cdot \frac{\frac{1}{a}}{-3}} \]

       3. associate-/l/ 75.8%

          \[\leadsto 0 - \sqrt{-3 \cdot \left(c \cdot a\right)} \cdot \color{blue}{\frac{1}{-3 \cdot a}} \]

       4. *-commutative 75.8%

          \[\leadsto 0 - \sqrt{-3 \cdot \left(c \cdot a\right)} \cdot \frac{1}{\color{blue}{a \cdot -3}} \]

       5. un-div-inv 75.9%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{a \cdot -3}} \]

       6. *-commutative 75.9%

          \[\leadsto 0 - \frac{\sqrt{-3 \cdot \color{blue}{\left(a \cdot c\right)}}}{a \cdot -3} \]

       7. associate-*r* 75.9%

          \[\leadsto 0 - \frac{\sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c}}}{a \cdot -3} \]

       8. *-commutative 75.9%

          \[\leadsto 0 - \frac{\sqrt{\color{blue}{\left(a \cdot -3\right)} \cdot c}}{a \cdot -3} \]

       9. associate-*l* 75.8%

          \[\leadsto 0 - \frac{\sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right)}}}{a \cdot -3} \]

   15. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{0 - \frac{\sqrt{a \cdot \left(-3 \cdot c\right)}}{a \cdot -3}} \]
   16. Step-by-step derivation

       1. neg-sub0 75.8%

          \[\leadsto \color{blue}{-\frac{\sqrt{a \cdot \left(-3 \cdot c\right)}}{a \cdot -3}} \]

       2. distribute-neg-frac2 75.8%

          \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(-3 \cdot c\right)}}{-a \cdot -3}} \]

       3. *-commutative 75.8%

          \[\leadsto \frac{\sqrt{a \cdot \color{blue}{\left(c \cdot -3\right)}}}{-a \cdot -3} \]

       4. distribute-rgt-neg-in 75.8%

          \[\leadsto \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{\color{blue}{a \cdot \left(--3\right)}} \]

       5. metadata-eval 75.8%

          \[\leadsto \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot \color{blue}{3}} \]

   17. Simplified 75.8%

       \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}} \]

   if 1.69999999999999988e-82 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 12.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 91.1%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-227}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-114)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.3e-227)
     (* 0.3333333333333333 (sqrt (/ (* c -3.0) a)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.3e-227) {
		tmp = 0.3333333333333333 * sqrt(((c * -3.0) / a));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.2d-114)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.3d-227) then
        tmp = 0.3333333333333333d0 * sqrt(((c * (-3.0d0)) / a))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-114) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.3e-227) {
		tmp = 0.3333333333333333 * Math.sqrt(((c * -3.0) / a));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-114:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.3e-227:
		tmp = 0.3333333333333333 * math.sqrt(((c * -3.0) / a))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-114)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.3e-227)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(Float64(c * -3.0) / a)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e-114)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.3e-227)
		tmp = 0.3333333333333333 * sqrt(((c * -3.0) / a));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-114], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-227], N[(0.3333333333333333 * N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.20000000000000036e-114

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 90.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 90.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -7.20000000000000036e-114 < b < 1.30000000000000006e-227

   1. Initial program 78.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 78.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 78.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 78.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 78.5%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 78.4%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 78.5%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 78.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 78.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 78.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around 0 38.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}} \]
   12. Step-by-step derivation

       1. rem-cube-cbrt 38.4%

          \[\leadsto 0.3333333333333333 \cdot \sqrt{\frac{c \cdot \color{blue}{-3}}{a}} \]

   13. Simplified 38.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}} \]

   if 1.30000000000000006e-227 < b

   1. Initial program 20.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 20.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 20.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 20.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 81.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-114}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-227}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{c \cdot -3}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 73.3%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 73.3%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 73.3%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 73.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 73.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 73.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 73.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 74.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 27.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b}{a} \cdot 2}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (* (/ b a) 2.0) -3.0) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = ((b / a) * 2.0) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = ((b / a) * 2.0d0) / (-3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = ((b / a) * 2.0) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = ((b / a) * 2.0) / -3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(Float64(b / a) * 2.0) / -3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = ((b / a) * 2.0) / -3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(N[(b / a), $MachinePrecision] * 2.0), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 79.3%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 79.2%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 79.2%

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. fma-undefine 79.2%

         \[\leadsto \frac{\left(b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}}\right) \cdot \frac{1}{a}}{-3} \]

      3. add-sqr-sqrt 54.5%

         \[\leadsto \frac{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right) \cdot \frac{1}{a}}{-3} \]

      4. hypot-define 60.2%

         \[\leadsto \frac{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a}}{-3} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \color{blue}{\left(-3 \cdot c\right)}}\right)\right) \cdot \frac{1}{a}}{-3} \]

   10. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-3 \cdot c\right)}\right)\right) \cdot \frac{1}{a}}{-3}} \]

   11. Taylor expanded in b around -inf 73.3%

       \[\leadsto \frac{\color{blue}{2 \cdot \frac{b}{a}}}{-3} \]
   12. Step-by-step derivation

       1. *-commutative 73.3%

          \[\leadsto \frac{\color{blue}{\frac{b}{a} \cdot 2}}{-3} \]

   13. Simplified 73.3%

       \[\leadsto \frac{\color{blue}{\frac{b}{a} \cdot 2}}{-3} \]

   if -3.999999999999988e-310 < b

   1. Initial program 27.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ b (* -1.5 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b / ((-1.5d0) * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b / (-1.5 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b / Float64(-1.5 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b / (-1.5 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b / N[(-1.5 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 79.3%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 79.2%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 79.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 79.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around -inf 73.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 73.2%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-*r/ 73.3%

          \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   13. Simplified 73.3%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
   14. Step-by-step derivation

       1. clear-num 73.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 73.2%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

       3. div-inv 73.3%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 73.3%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   15. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 27.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 79.3%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 79.2%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 79.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 79.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around -inf 73.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 73.2%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-*r/ 73.3%

          \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   13. Simplified 73.3%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 27.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 79.3%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

      2. div-inv 79.2%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      2. *-commutative 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

      3. associate-/r* 79.2%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   8. Simplified 79.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 79.0%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      2. pow3 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      3. *-commutative 79.1%

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   10. Applied egg-rr 79.1%

       \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   11. Taylor expanded in b around -inf 73.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 73.2%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-*r/ 73.3%

          \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   13. Simplified 73.3%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 27.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-*l* 27.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 56.7%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. clear-num 56.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b}}}} \]

      2. inv-pow 56.3%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b}}\right)}^{-1}} \]

      3. times-frac 56.4%

         \[\leadsto {\color{blue}{\left(\frac{3}{-1.5} \cdot \frac{a}{\frac{a \cdot c}{b}}\right)}}^{-1} \]

      4. metadata-eval 56.4%

         \[\leadsto {\left(\color{blue}{-2} \cdot \frac{a}{\frac{a \cdot c}{b}}\right)}^{-1} \]

      5. associate-/l* 60.9%

         \[\leadsto {\left(-2 \cdot \frac{a}{\color{blue}{a \cdot \frac{c}{b}}}\right)}^{-1} \]

   7. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{{\left(-2 \cdot \frac{a}{a \cdot \frac{c}{b}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 60.9%

         \[\leadsto \color{blue}{\frac{1}{-2 \cdot \frac{a}{a \cdot \frac{c}{b}}}} \]

      2. associate-/r* 72.0%

         \[\leadsto \frac{1}{-2 \cdot \color{blue}{\frac{\frac{a}{a}}{\frac{c}{b}}}} \]

      3. *-inverses 72.0%

         \[\leadsto \frac{1}{-2 \cdot \frac{\color{blue}{1}}{\frac{c}{b}}} \]

   9. Simplified 72.0%

      \[\leadsto \color{blue}{\frac{1}{-2 \cdot \frac{1}{\frac{c}{b}}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 72.0%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{-2 \cdot \frac{1}{\frac{c}{b}}}} \]

       2. associate-/r* 72.0%

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{-2}}{\frac{1}{\frac{c}{b}}}} \]

       3. metadata-eval 72.0%

          \[\leadsto 1 \cdot \frac{\color{blue}{-0.5}}{\frac{1}{\frac{c}{b}}} \]

       4. clear-num 71.9%

          \[\leadsto 1 \cdot \frac{-0.5}{\color{blue}{\frac{b}{c}}} \]

   11. Applied egg-rr 71.9%

       \[\leadsto \color{blue}{1 \cdot \frac{-0.5}{\frac{b}{c}}} \]
   12. Step-by-step derivation

       1. *-lft-identity 71.9%

          \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

       2. associate-/r/ 72.2%

          \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

   13. Simplified 72.2%

       \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. sqr-neg 54.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. sqr-neg 54.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. associate-*l* 54.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 54.1%

      \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-3 \cdot a}} \]

   2. div-inv 54.0%

      \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

6. Applied egg-rr 54.0%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
7. Step-by-step derivation

   1. *-commutative 54.0%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

   2. *-commutative 54.0%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right)}\right) \cdot \frac{1}{a \cdot -3} \]

   3. associate-/r* 54.0%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

8. Simplified 54.0%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
9. Step-by-step derivation

   1. add-cube-cbrt 53.8%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot -3} \cdot \sqrt[3]{c \cdot -3}\right) \cdot \sqrt[3]{c \cdot -3}\right)}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   2. pow3 53.9%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{c \cdot -3}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

   3. *-commutative 53.9%

      \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot {\left(\sqrt[3]{\color{blue}{-3 \cdot c}}\right)}^{3}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

10. Applied egg-rr 53.9%

    \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{{\left(\sqrt[3]{-3 \cdot c}\right)}^{3}}\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

11. Taylor expanded in b around -inf 39.0%

    \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 39.0%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

    2. *-commutative 39.0%

       \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

    3. associate-*r/ 39.0%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

13. Simplified 39.0%

    \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))